Semi-Square and Sesquiquadrate: The 45° and 135° Aspects in a Birth Chart

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The semi-square and the sesquiquadrate are two minor aspects built from the square. A semi-square joins two planets 45° apart, half a square. A sesquiquadrate joins two planets 135° apart, a square and a half. This guide sets out the geometry, explains why you have to work these two out in degrees rather than signs, gives the band it uses to decide whether a pair counts, and checks real separations from three example charts, including one that sits too close to the edge to call.

Educational minor aspects, not calculator features. Astrologist.com explains the semi-square and sesquiquadrate as an educational extra. The free birth chart calculates and draws only the five major aspects (conjunction, sextile, square, trine and opposition); it does not list, flag or draw semi-squares or sesquiquadrates. It does print every planet's sign, degree and minutes, so you can measure these two angles yourself with the method below. Ask the Astrologist is not given your planets' degrees and does not detect these aspects; it can discuss the idea, or a separation you have measured and typed in. The figures on this page were worked out offline with the same planetary routines as the free chart. Nothing here changes how the calculator works.

The geometry: eighths of the circle

Divide the 360° circle by eight and you get 45°. Count in eighths and you meet four angles in turn: 45°, 90°, 135° and 180°. Two of those are major aspects, the square and the opposition. The other two are the minor ones this page covers:

Because they come from the same division as the square and the opposition, astrologers usually group all four together; in harmonic language this is called the eighth-harmonic family. The names come from Latin: semi is half, and sesqui means one and a half, so a sesquiquadrate is "one and a half quadrates (squares)". You will also meet octile for the semi-square, and sesquisquare or trioctile for the sesquiquadrate. These pages use semi-square and sesquiquadrate throughout.

One useful consequence of the geometry: 45° + 135° = 180°. If two planets are in opposition, a third planet sitting 45° from one of them is automatically about 135° from the other. That is arithmetic, not an extra rule, and it is a quick way to spot a sesquiquadrate once you have found a semi-square near an opposition.

No sign-pair shortcut

Major aspects line up with the signs: a square usually joins signs three apart, a trine signs four apart, and the quincunx page can say that a 150° contact usually links signs five apart. The semi-square and sesquiquadrate cannot be read that way. 45° is one and a half signs and 135° is four and a half, so neither angle is a whole number of signs.

In practice the same two signs can hold a semi-square or nothing at all:

• 2° Aries to 16° Taurus is 44°: one degree from exact, inside this page's band.
• 2° Aries to 20° Taurus is 48°: three degrees from exact, outside it.
• 26° Aries to 10° Gemini is 44°: a semi-square even though the signs are two apart, not neighbours.

So there is no "in-sign" or "out-of-sign" label for these aspects and no table of sign pairs to look up. The only reliable test is the degree count. Out-of-sign aspects explains the same degree-versus-sign question for the major aspects, where it does matter.

This guide's working band: 2° from exact

Every aspect needs a tolerance, the distance from exact that still counts. On this page a semi-square is any separation from 43° to 47°, and a sesquiquadrate any separation from 133° to 137°, both limits included: within 2° of exact, measured in zodiac longitude. The same 2° applies whichever planets are involved. There is no extra allowance for the Sun or the Moon and no wider band for "personal" planets.

This is the guide's own educational convention, chosen so that every example can be reproduced. It is not a calculator setting, and it does not change the site's major-aspect orbs (8° for most aspects, 6° for the sextile), which are explained on aspect orbs. Published practice is wider than one number:

• The TimePassages desktop software's default natal settings use 2° for both the semi-square and the sesquiquadrate (and widen orbs by half when the Sun or Moon is involved, which this page does not do).
• Astrodienst's Astrowiki article on the sesquiquadrate gives an orb of 2 to 3 degrees.
• Astrotheme's natal settings use 1.5° for the semi-square and 2.5° for the sesquiquadrate.

A tighter or wider band will find fewer or more of these aspects in the same chart. Neither result is a mistake; it is a different convention, and it is worth knowing which one a source uses before comparing.

Checking a pair by hand

You need the sign, degree and minutes of each planet from the planet table under your free chart.

1. Convert each position to a longitude from 0° Aries. Add the start of the sign (Aries 0, Taurus 30, Gemini 60, Cancer 90, Leo 120, Virgo 150, Libra 180, Scorpio 210, Sagittarius 240, Capricorn 270, Aquarius 300, Pisces 330) to the degree, plus the minutes divided by 60.
2. Find the shorter separation. Subtract one longitude from the other and drop the minus sign. If the result is more than 180°, take it away from 360°.
3. Compare with 45° and 135°. If the separation is within 2° of either (43° to 47°, or 133° to 137°), the pair is inside this guide's band for that aspect.

Only longitude is used. Declination, latitude, house-space (mundane) measures and whether the aspect is applying or separating are all outside this method.

The ~2′ safeguard. The chart prints whole minutes and drops the seconds, so a hand result can be off by a minute or two. If your answer is within about 2′ of the band edge (for example 2°00′ or 1°59′ from exact), treat the pair as boundary-sensitive: the printed figures cannot settle whether it is just inside or just outside, and it is better to say so than to pick an answer.

Which bodies. This guide uses the ten planets from the Sun to Pluto. The lunar Nodes, Chiron, Lilith and the chart angles are outside its method, even though some astrologers apply minor aspects to them. Contacts between two of the outer planets (Uranus, Neptune, Pluto) last for years and are shared by everyone born in that stretch, so these pages do not read them as personal traits.

Sun, Mercury and Venus

Mercury and Venus never stray far from the Sun, which rules out some of these aspects between the three before you look at a chart. Using the site's own measured maxima (see which Sun, Mercury and Venus aspects are possible), rechecked for this page over daily noon charts from 1900 to 2099:

So under this 2° band the Sun and Mercury never form either aspect, and no pair of the three ever forms a sesquiquadrate. These figures are the site's measured reference geometry from noon samples, not a mathematical proof for every possible moment, but the margins are wide. Ruben's chart below has a Sun-Venus semi-square, one of the few contacts this pair can make.

Worked examples

Three fictional charts recur across Astrologist.com guides. Their positions were recalculated for this page with the production chart routines; the hand figures below use the printed (truncated) minutes, exactly as a reader would.

Lotte: a clear sesquiquadrate

Lotte, Utrecht, 27 August 1991, 06:55. Mercury is at 24°18′ Leo (retrograde) and Uranus at 10°03′ Capricorn (retrograde).

• Longitudes: 120 + 24°18′ = 144°18′ (144.30°) and 270 + 10°03′ = 280°03′ (280.05°).
• Difference: 135°45′. That is under 180°, so it is the separation.
• From exact 135°: 0°45′, inside the band, well clear of the edge.

Recalculated at every minute of her birth date, the separation stays between about 135.6° and 136.1°, so the result does not depend on her birth time. Her chart has two more sesquiquadrates to Uranus, from Venus (1°31′ from exact) and Jupiter (1°29′), also inside the band all day.

Ruben: a semi-square

Ruben, London, 5 January 1993, 12:00 GMT. Venus is at 1°39′ Pisces and Uranus at 17°56′ Capricorn.

• Longitudes: 331°39′ (331.65°) and 287°56′ (287.93°).
• Separation: 43°43′, which is 1°17′ from exact 45°: inside the band.

Across the whole London date the pair runs from about 43.2° to 44.2°, so this semi-square holds at any time of day. Ruben's Sun at 15°10′ Capricorn and Venus are 46°29′ apart, 1°29′ from exact, a second semi-square that also holds all day.

Ruben: a boundary-sensitive case

Venus at 1°39′ Pisces (331°39′) and Mars at 18°38′ Cancer, retrograde (108°38′):

• Difference 223°01′, more than 180°, so the separation is 360° − 223°01′ = 136°59′.
• From exact 135°: 1°59′, one minute inside the 2°00′ edge.

That is inside the ~2′ safeguard. The unrounded calculation puts the pair at about 1.99° from exact at noon, just inside, but the printed minutes cannot show that reliably, so on this page the pair is boundary-sensitive, not a clean sesquiquadrate. The time matters as well: over his birth date the separation shrinks from about 137.7° at midnight to about 136.2° late in the evening, and it only comes inside 2° from about 11:47. A Ruben with no known birth time could not claim it either way.

Noor: a non-Moon pair that depends on the time

Noor, Amsterdam, 2 December 1992, 21:30. The Sun is at 10°55′ Sagittarius (250°55′) and Mars at 27°31′ Cancer, retrograde (117°31′). The separation is 133°24′, 1°36′ from exact 135°: inside the band at her stated time.

Neither planet is the Moon, yet the result is not stable across the day. The Sun moves forward about a degree a day while retrograde Mars slips back, and the separation only reaches 133° from about 12:23. Born earlier that day, the same person would not have this aspect under a 2° band. Noor also has Jupiter at 9°59′ Libra and Pluto at 23°34′ Scorpio, 43°35′ apart: a semi-square 1°25′ from exact that holds all day.

None of these contacts appears on the free chart. No pair above is within a major-aspect orb, so the wheel draws no line between them.

Without a birth time

The Moon moves roughly 12° to 15° a day, so a semi-square or sesquiquadrate to the Moon cannot be claimed without a reliable birth time, unless the pair has been checked minute by minute across the whole possible birth window and stays inside the band throughout. The chart's internal noon placeholder for an unknown time is a reference point, not your birth moment.

Do not assume that a contact between two slower planets is safe either. Noor's Sun-Mars pair above changes its answer in the middle of the day, and Ruben's Venus-Mars pair crosses the edge at about 11:47. Lotte's Mercury-Uranus and Ruben's Venus-Uranus are stable only because they were checked at every minute of the date, not because slow pairs are stable by rule. Your birth chart without a birth time covers what an untimed chart can and cannot show.

How these two aspects are usually read

Astrologers who use them generally treat the semi-square and sesquiquadrate as relatives of the square: contacts that are read in terms of tension, friction or a need for adjustment between the two planets, usually described as subtler and less obvious than a square. Some writers describe the semi-square as having an irritating quality, a prompt to adjust an attitude or approach, and give the sesquiquadrate a similar square-like character. These are traditional and modern readings, stated as tendencies, not as predictions of conflict, crisis or events, and they say nothing fixed about what a person will do.

As with any aspect, the planets set the subject: a Mercury-Uranus sesquiquadrate would be read as friction between habitual thinking and the urge to break pattern, a Sun-Venus semi-square as a small mismatch between self-expression and what feels pleasant. Planet meanings covers what each planet stands for. Many astrologers give these aspects less weight than the five major aspects, and they are usually read in support of the main picture rather than on their own.

A note on history

The 135° sesquiquadrate is one of three angles Johannes Kepler proposed as new aspects in the early seventeenth century, alongside the 72° quintile and the 144° bi-quintile (see quintile and bi-quintile); he set them out most fully in Harmonices Mundi (The Harmony of the World). The semi-square was not among Kepler's three. William Lilly, writing Christian Astrology in 1647, gave instructions for finding Kepler's new aspects and also the 45° aspect, among others, as summarised in the Skyscript article "Johannes Kepler: Music of the Spheres". The same article notes that in Lilly's worked examples the square-based aspects came out as "slightly harmful" and the quintile-based ones as "mildly beneficial", a split that later textbooks kept; modern readings, as above, frame them as tendencies rather than verdicts.

How these aspects fit with other topics

Minor aspects gives the general picture of the smaller angles, and the quincunx covers the 150° contact. When you count a chart's aspects, semi-squares and sesquiquadrates are not included: how many aspects a birth chart has counts the five major aspects only. A planet whose only contacts are semi-squares or sesquiquadrates is still unaspected under the five-aspect test these guides use; unaspected or not: borderline cases explains why some astrologers would disagree. For the other minor family Kepler introduced, built on fifths of the circle, see the quintile and bi-quintile. The methodology page lists what the calculator does compute.

Related articles

• Quintile and Bi-Quintile: The 72° and 144° Aspects
• Quincunx (Inconjunct) Aspect
• Minor Aspects in Astrology
• Major Aspects in Astrology
• Aspect Orbs in Astrology
• Out-of-Sign Aspects
• Sun, Mercury and Venus: Which Aspects Are Possible?
• Unaspected or Not: Borderline Cases

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